Goals from Philippe Coutinho and Daniel Sturridge gave the visitors a 2-0 lead at half time, but Koeman’s men produced a stunning second-half comeback.
Sadio Mané, who had earlier missed a penalty, got Saints back in it, before Graziano Pellè’s excellent strike levelled the game. Mané then won it for the hosts four minutes from time to spark wild celebrations at St Mary’s.
Speaking to youtube.com/southamptonfc
afterwards, Koeman said: “We were lucky in half time to only be two-nil down. We spoke about that.
“We needed to show the belief and the spirit in coming back into the game and to make it one-two, because then it’s a really open game.
“That happened. Great spirit. A great second half. The substitutes, Sadio and Victor [Wanyama], had a big impact in the game in the second half.
“It was well deserved, a great game and it was great support from the fans. This is what football is about.
“I’m proud of the team and the fans. It’s easy to tell everybody because you get three points, but even with the two-two or the one-two, how we came back in the game was amazing.”
And Koeman admits it probably wasn’t the best idea for him to sprint down the touchline after Mané’s winner, but says the emotion of the game got the better of him.
“I don’t do that anymore really, especially if I think about my Achilles operation! It wasn’t best, but I was emotional,” Koeman added.
“At two-nil down against Liverpool, they let us live, the reaction of the players and winning the game makes it special.
“I’m very proud of the team.”